We begin by examining the relation between good simplicial
objects and proper simplicial objects. The slogan “good implies proper” seems to be folklore in some contexts.
This is implicit in the proof of Lemma A–5 of [1]} and a
proof in the context of simplicial spaces is given in Corollary 2.4 (b) of [2]},
while a similar statement is made in Remark 22.4.3 of [3]}.
We will give a proof here in a reasonably general setting under the assumption that some additional axioms hold,
one of which is a generalization of Lillig's union theorem on cofibrations [4]}.
